CCSS Alignments

Hot Dogs for a Picnic, 3-5

Exemplars math material is an effective way to incorporate both the Common CoreStandards for Mathematical Practice and the Standards for Mathematical Content into your classroom. Learn how our performance tasks and assessment rubrics support CCSSM.

Task Alignments to the Standards for Mathematical Practice

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimationand other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Task Alignments to the Standards for Mathematical Content

Grade 3

3. Use multiplication and division within 100 to solve word problems in
situations involving equal groups, arrays, and measurement quantities,
e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem.

Multiply and divide within 100.

7. Fluently multiply and divide within 100, using strategies such as the
relationship between multiplication and division (e.g., knowing that 8 x
5 = 40, one knows 40 / 5 = 8) or properties of operations. By the end
of Grade 3, know from memory all products of two one-digit numbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

8. Solve two-step word problems using the four operations. Represent these
problems using equations with a letter standing for the unknown
quantity. Assess the reasonableness of answers using mental computation
and estimation strategies including rounding.

Grade 4

2. Multiply or divide to solve word problems involving multiplicative
comparison, e.g., by using drawings and equations with a symbol for the
unknown number to represent the problem, distinguishing multiplicative
comparison from additive comparison.

3. Solve multistep word problems posed with whole numbers and having
whole-number answers using the four operations, including problems in
which remainders must be interpreted. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the
reasonableness of answers using mental computation and estimation
strategies including rounding.

Gain familiarity with factors and multiples.

4. Find all factor pairs for a whole number in the range 1-100. Recognize
that a whole number is a multiple of each of its factors. Determine
whether a given whole number in the range 1-100 is a multiple of a given
one-digit number. Determine whether a given whole number in the range
1-100 is prime or composite.

2. Read and write multi-digit whole numbers using base-ten numerals, number
names, and expanded form. Compare two multi-digit numbers based on
meanings of the digits in each place, using >, =, and <. symbols
to record the results of comparisons.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

5. Multiply a whole number of up to four digits by a one-digit whole
number, and multiply two two-digit numbers, using strategies based on
place value and the properties of operations. Illustrate and explain the
calculation by using equations, rectangular arrays, and/or area models.

Several years ago, Connie Witt tried to implement a standards-based curriculum and grading system into her school district, Los Alamos. At that time, she received little feedback or enthusiasm from her colleagues. Nevertheless, she pushed on. Determined to turn things around, Connie purchased Exemplars with her own money and began to apply basic standards-based skills...